Suppose you have two variables x and y, then you have a function $$u(x,y)$$ Then if you have a function $$f(x,y,u(x,y))$$ what does it mean to take the partial derivative of f with respect to u?
More specifically, if you have $$f(x,y,u(x,y))=F(g(x,y,u(x,y))$$ for some functions F and g why can't you say that $$\frac{\partial f(x,y,u(x,y))}{\partial u} = F'(g(x,y,u(x,y))\frac{\partial g(x,y,u(x,y))}{\partial u}$$?
